‎Let $S$ be an inverse semigroup with the set of idempotents $E$‎.We prove that the semigroup algebra $\ell^{1}(S)$ is always‎‎$2n$-weakly module amenable as an $\ell^{1}(E)$-module‎, ‎for any‎‎$n\in \mathbb{N}$‎, ‎where $E$ acts on $S$ trivially from the left‎‎and by multiplication from the right‎. ‎Our proof is based on a common fixed point property for semigroups‎. Manuscript profile

‎Let $ \mathcal{A} $ be a unital Banach algebra‎, ‎$ w\in \mathcal{A}$‎, ‎and $ \gamma‎ : ‎\mathcal{A} \to \mathcal{A} $ is a continuous linear map‎. ‎We show that $\gamma$ satisfies $a\gamma(b)=\gamma(w)$ ($\gamma(a)b=\gamma(w)$) whenever $a,b\in \mathcal{A}$ with $ab=w$ and $w$ is a left (right) separating point in $\mathcal{A}$ if and only if $\gamma$ is a right (left) centralizer‎. ‎Also‎, ‎we prove that $\gamma$ satisfies $a\gamma(b)=\gamma(a)b=\gamma(w)$ whenever $a,b\in \mathcal{A}$ with $ab=w$ and $w$ is a left or right separating point in $\mathcal{A}$ if and only if $\gamma$ is a centralizer‎. ‎We also provide some applications of the obtained results for characterization of a continuous linear map $\gamma:\mathcal{A}\rightarrow \mathcal{A}$ on a unital Banach $*$-algebra $\mathcal{A}$ satisfying $a\gamma(b)^{*}=\gamma(w^{*})^{*}$ ($\gamma(a)^{*}b=\gamma(w^{*})^{*}$) whenever $a,b\in \mathcal{A}$ with $ab^{*}=w$ ($a^{*}b=w$) and $w$ is a left (right) separating point‎, ‎or $\gamma$ satisfying $a\gamma(b)^{*}=\gamma(c)^{*}d=\gamma(w^{*})^{*}$ whenever $a,b,c,d\in \mathcal{A}$ with $ab^{*}=c^{*}d =w$ and $w$ is a left or right separating point‎. Manuscript profile